1 Radian Measures. Exercise 1. A --- 1Vc. MAT Worksheet 10 Sections 7.1, 8.1 Name:

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1 MAT Radian Measures Consider he following figure. corresponding o he angle 0. Exercise 1 The shaded porion of he circle is called he secor of he circle K V 1. Suppose know he radian measure of he angle 0 and he arc lengh s. Wha is he radius r of he circle, in erms of s and 0? S m 2. Suppose know he radius r arid of he circle spanned by 0? he angle 6. Wha is he arc lengh around he fracion 3. Wha is he area A of he shaded secor? (Hin: his par conains he fracior(f he whole area of he circle.) A --- 1Vc 7-1

2 MAT 012 SS218 Exercise 2 We will use our knowledge of rigonomery o undersand how a (very simplified) bicycle func ions. 1. Suppose ha pedal a a rae of 120 revoluions per minue. (A revoluion is one full roaion of 3600 around he axis.) Wha is he angular velociy of he pedal sysem, expressed in radians per second? zv c&\ays (&Osecco-) 2. The pedals are direcly conneced o he rear wheel, so he jg]arvelocj of he wheels is he same as he angular speed of he pedals (which you found in par (1)). Given ha he rear ire has a diameer of 69cm. wha is he linear speed of a poin on he rear wheel? - S - r 3. The fron gear sprocke is conneced o he pedals by a chain. n oher words, he linear disance covered by one pedal is he same as ile linear disance covered by any poin on audhe?iiiiutfie fron gear sprockdf]zndlië.wa is he angular speed of he fron gear sprocke? \Z JJz z -i; - 2

3 in radians. beween degrees and radians, o complee he fouowing able where each of he degree measures is Use wha you know abou he rigonomeric funcions for hese angles, and he relaionship Exercise 3 3 Ac Osn anoher imporan rigonomeric ideniy. 2. Divide boh sides of he original equaion above by cos2(o). Wha do you ge? This is cscth imporan rigonomeric ideniy. 1. Divide boh sides of he equaion above by sin2(o). Wha do you ge? This is anoher We proved his using he pyhagorean heorem and he uni circle. sin2(o) + cos2(o) = 1. Raafl ha for any angle B beween 0 and 2ir, Exercise 4 2 Trigonomeric funcions of real numbers MAT 012 SS218

4 4 Wa-v\ W, 1. Suppose is beween i and w. Draw a picure and use i o explain why cos() = cos( ). Exercise 5 C /- 1 locaion of on he uni circle, o find cos and an i. Exercise 6 cr A - ycqc5hc V- L Ca8Afl0L\<S * fvoüix. è W Q- C- 1-c?cc\a@klr\ \3 ir-\-oiv\& CO LS 4-VQ- )c Ca 0c- (ic1 Workshee 10 Secions 7.1, 8.1 Name:?vAT 012 SS Do you hink cos() = cos( ) for any real number? Why or why rio? 3. Suppose is beween i- and in-. Draw a picure and use i o explain why sinu) = sin( ). 1. Suppose slu = and < < f Use he rigonomeric ideniies you how, and he 4 = Th u&s 05 CJC 0-a + 4. Do you hink sin() = sin( i) for any real number? Why or why no? is

5 U (ac-fl rn n,jc&c cos =? sch 7 csc= 2. Suppose sec = 3 and 0 < < ir. Use ile rigonomeric ideniies you know, and he locaion MAT of on he uni circle, o find an and csc , , 17-22, 25, 27-49, Addiional Recommended Exercises 5. Simplify he expression ak-c co s-4cax-. c_- e L4 Li C C059 S =1 answers consisen? in he previous par for 8 using rigonomeric ideniies. Wha is cos(o)? Are your wo 4. Now, make he subsiuion = cos(o), supposing 8 is beween 0 and ir. Solve he equaion 3. Suppose > 0 arid = 1. Solve for using arihmeic operaions. os ki 4 1:: 2 sn = &= cn - (-p5 5\fl 3c csc sin i sin cos2 co cscz / sn3 - i-si CS C CO5Zk 9 -c7 yirc 2 Q, -3